1. Field of the Invention
The present invention relates to computerized simulation of hydrocarbon reservoirs in the earth, and in particular to simulation of flow profiles along wells in a reservoir.
2. Description of the Related Art
Well models have played an important role in numerical reservoir simulation. Well models have been used to calculate oil, water and gas production rates from wells in an oil and gas reservoirs. If the well production rate is known, they are used to calculate the flow profile along the perforated interval of the well. With the increasing capabilities for measuring flow rates along the perforated intervals of a well, a proper numerical well model is necessary to compute the correct flow profile to match the measurements.
It is well known that simple well models such as explicit or semi implicit models could be adequate if all reservoir layers communicated vertically. For these models, well production rate was allocated to the perforations in proportion to the layer productivity indices (or total mobility). Therefore, the calculations were simple. The resulting coefficient matrix for the unknowns remained unchanged. Specifically, the coefficient matrix maintained a regular sparse structure. Therefore, any such sparse matrix solver could be used to solve the linear system for the grid block pressures and saturations for every time step.
However, for highly heterogeneous reservoirs with some vertically non-communicating layers, the above-mentioned well models did not produce the correct physical solution. Instead, they produced incorrect flow profiles and in some occasions caused simulator convergence problems.
With the increasing sophistication of reservoir models, the number of vertical layers has come to be in the order of hundreds to represent reservoir heterogeneity. Fully implicit, fully coupled well models with simultaneous solution of reservoir and well equations have been necessary to correctly simulate the flow profiles along the well and also necessary for the numerical stability of the reservoir simulation. In order to solve the fully coupled system, generally well equations were eliminated first. This created an unstructured coefficient matrix for the reservoir unknowns to be solved. Solutions of this type of matrices required specialized solvers with specialized preconditioners. For well with many completions and many wells in a simulation model, this method has become computationally expensive in terms of processor time.